For twice smooth functions, the symmetry of the matrix of second partial derivatives is automatic and can be seen as the symmetry of the Jacobian matrix of the gradient mapping. For nonsmooth functions, possibly even extended-real-valued, the gradient mapping can be replaced by a subgradient mapping, and generalized second derivative objects can then be introduced through graphical differentiat...

An analysis is made of reality conditions within the context of noncommutative geometry. We show that if a covariant derivative satisfies a given left Leibniz rule then a right Leibniz rule is equivalent to the reality condition. We show also that the matrix which determines the reality condition must satisfy the Yang-Baxter condition if the extension of the covariant derivative to tensor produ...

This paper surveys and compares some recent approaches to stochastic infinite-dimensional geometry on the space Γ of configurations (i. e. locally finite subsets) of a Riemannian manifold M under Poisson measures. In particular, different approaches to Bochner– Weitzenböck formulas are considered. A unitary transform is also introduced by mapping functions of n configuration points to their mul...

Aim: To investigate the effect of Emdogain gel (EMD) in pulpotomized primary molars and its clinical and radiographic outcomes. Methods and Materials: In this   study, 18 lower second primary molars of nine children were treated by   pulpotomy. The teeth were randomly assigned to the EMD (experimental) and Formocresol (control) groups in each patient (split mouth). Following removal of the coro...

Enamel matrix derivative Emdogain (EMD) is widely used in periodontal treatment in spite of the fact that its effect on the developing embryo has not been elucidated. The aim of this study was to investigate the teratogenic effect of EMD on the rat embryo neural crest cells. The neural crest is a unique population of cells that migrates from the dorsal neural tube along defined pathways and pro...

Automatic differentiation (AD) is a way to accurately and efficiently compute derivatives of a function written in computer codes. We describe the procedures necessary to apply the AD method to the multiphase, multicomponent, nonisothermal flow simulator TOUGH2. In particular, we apply the AD method to the ECO2 module of the TOUGH2 code to explore a scheme for efficiently calculating the Jacobi...